The wording “graph find the inequality represented by the graph” points to a shaded half-plane bounded by a straight line. The boundary line gives the associated linear equation, and the shaded side determines whether the inequality is “greater than” or “less than.”
The shaded solution set corresponds to \[ y \ge -\frac{1}{2}x + 2. \]
Boundary line and inequality symbol
The boundary is the line itself, written in slope-intercept form as \(y = mx + b\). A solid boundary indicates that points on the line are included, so the symbol is \(\ge\) or \(\le\). A dashed boundary indicates exclusion, so the symbol is \(>\) or \(<\).
| Graph feature | Algebraic meaning | Symbol choice |
|---|---|---|
| Solid boundary line | Boundary points satisfy the inequality | \(\ge\) or \(\le\) |
| Dashed boundary line | Boundary points do not satisfy the inequality | \(>\) or \(<\) |
| Shading above the line | \(y\) values larger than the boundary line at each \(x\) | \(\ge\) or \(>\) |
| Shading below the line | \(y\) values smaller than the boundary line at each \(x\) | \(\le\) or \(<\) |
Computing the boundary equation for the illustrated line
Two labeled points on the line are \((0,2)\) and \((4,0)\). The slope is \[ m=\frac{0-2}{4-0}=-\frac{2}{4}=-\frac{1}{2}. \] The \(y\)-intercept is \(b=2\) because the line crosses the \(y\)-axis at \((0,2)\). Therefore the boundary equation is \[ y=-\frac{1}{2}x+2. \]
The shaded region lies above the line, so \(y\) is greater than or equal to the line’s value at each \(x\). The solid boundary confirms inclusion, giving \[ y \ge -\frac{1}{2}x + 2. \]
Consistency check with a test point
A test point not on the boundary, such as \((0,0)\), verifies the shading direction. Substitution into the inequality gives \[ 0 \ge -\frac{1}{2}\cdot 0 + 2 \;\;\Longrightarrow\;\; 0 \ge 2, \] which is false, so \((0,0)\) must be outside the shaded region. The displayed shading excludes \((0,0)\) and includes points like \((0,3)\), consistent with the inequality.
Common pitfalls
Confusion between shading direction and the inequality symbol is common when the line slopes downward; the shading still corresponds to “above” versus “below” in the vertical sense. A second frequent error is mixing dashed and solid boundaries: strict inequalities \(>\) and \(<\) require a dashed line, while inclusive inequalities \(\ge\) and \(\le\) require a solid line.